1. Reasoning in Psychology Using Statistics
2015

2. Announcements Quiz 8 due Fri. Apr 17
Includes both correlation and regression
Final Project due date Wed. April 29th (you should get your cases assigned to you in labs today)
Announcements

3. Exam(s) 3 Lecture Exam 3 Lab Exam 3 Combined Exam 3
Mean 53.6 (53.6/75 = 71.4%)
Lab Exam 3
Mean 61.0 (61.0/75 = 81.3%)
Combined Exam 3
Mean (116.1/150 = 77.4%)
Exam(s) 3

4. Regression procedures can be used to predict the response variable based on the explanatory variable(s)
Suppose that you notice that the more you study for an exam, the better your score typically is.
This suggests that there is a relationship between the variables.
You can use this relationship to predict test performance base on study time.
study time
test performance
115 mins
15 mins
Regression

5. Decision tree Regression
Describing the nature of the relationship between variables for the purposes of prediction
Two variables
Relationship between variables
Quantitative variables
Making predictions based on form of the relationship
Decision tree

6. For correlation: “it doesn’t matter which variable goes on the X-axis or the Y-axis”
The variable that you are predicting (response variable) goes on the Y-axis
Predicted variable
For regression this is NOT the case Y X 1 2 3 4 5 6
Quiz performance
Predictor variable
The variable that you are making the prediction based on (explanatory variable) goes on the X-axis
Hours of study
Regression

7. Regression For correlation: “Imagine a line through the points”
But there are lots of possible lines Y X 1 2 3 4 5 6
One line is the “best fitting line”
Today: learn how to compute the equation corresponding to this “best fitting line”
Quiz performance
Hours of study
Regression

8. Regression A brief review of geometry Y = (X)(slope) + (intercept) 2.0
Y = intercept, when X = 0 Y X 1 2 3 4 5 6
Y = (X)(slope) + (intercept)
2.0
Regression

9. Regression A brief review of geometry Y = (X)(slope) + (intercept) 0.51 2 3 4 5 6
Y = (X)(slope) + (intercept)
0.5
2.0 1 2
Change in Y
Change in X
= slope
Regression

10. Regression A brief review of geometry Y = (X)(slope) + (intercept)1 2 3 4 5 6
Y = (X)(slope) + (intercept)
Y = (X)(0.5) + 2.0
In regression analysis this line (or the equation that describes it) represents our predicted values of Y given particular values of X
Regression

11. Regression A brief review of geometry Consider a perfect correlation
X = 5
Y = ? Y X 1 2 3 4 5 6
Y = (X)(0.5) + (2.0)
Y = (5)(0.5) + (2.0)
Y = = 4.5
4.5
Can make specific predictions about Y based on X
Regression

12. Regression Consider a less than perfect correlation
The line still represents the predicted values of Y given X
X = 5
Y = ? Y X 1 2 3 4 5 6
Y = (X)(0.5) + (2.0)
Y = (5)(0.5) + (2.0)
Y = = 4.5
4.5
Regression

13. The “best fitting line” is the one that minimizes the differences (error or residuals) between the predicted scores (the line) and the actual scores (the points) Y X 1 2 3 4 5 6
Rather than compare the errors from different lines and picking the best, we will directly compute the equation for the best fitting line
Regression

14. Example Using the dataset from our correlation lecture Y X
Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Compute the regression equation predicting exam score with study time. Y X 1 2 3 4 5 6 X Y A B C D E
Example

15. Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Compute the regression equation predicting exam score with study time. X Y A
2.4
5.76
2.0
4.0
4.8 B
-2.6
6.76
-2.0
4.0
5.2 C
1.4
1.96
2.0
4.0
2.8 D
-0.6
0.36
0.0
0.0
0.0 E
-0.6
0.36
-2.0
4.0
1.2
mean
3.6
4.0
0.0
15.20
SSX
0.0
16.0
SSY
14.0 SP
Example

16. Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Compute the regression equation predicting exam score with study time. X Y A B C D E SP
mean
3.6
4.0
15.20
SSX
16.0
SSY
14.0
Example

17. Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Compute the regression equation predicting exam score with study time. X Y A B C
4.0 D E SP
mean
3.6
15.20
SSX
16.0
SSY
14.0
Example

18. Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Compute the regression equation predicting exam score with study time. Y X 1 2 3 4 5 6 X Y A B C D E
mean
3.6
4.0
Example

19. The two means will be on the line
Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Compute the regression equation predicting exam score with study time. Y X 1 2 3 4 5 6 X Y
The two means will be on the line A B C D E
mean
3.6
4.0
Example

20. Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Compute the regression equation predicting exam score with study time. Y X 1 2 3 4 5 6 X Y A B C
Hypothesis testing
on each of these D E
mean
3.6
4.0
Example

21. Hypothesis testing with Regression
SPSS Regression output gives you a lot of stuff
Hypothesis testing with Regression

22. Hypothesis testing with Regression
SPSS Regression output gives you a lot of stuff
Make sure you put the variables in the correct role
Hypothesis testing with Regression

23. Hypothesis testing with Regression
SPSS Regression output gives you a lot of stuff
Unstandardized coefficients
“(Constant)” = intercept
Variable name = slope
These t-tests test hypotheses
H0: Intercept (constant) = 0
H0: Slope = 0
Hypothesis testing with Regression

24. Measures of Error in Regression
The linear equation isn’t the whole thing
Also need a measure of error
Y = X(.5) + (2.0) + error
Y = X(.5) + (2.0) + error
Same line, but different relationships (strength difference) Y X 1 2 3 4 5 6 Y X 1 2 3 4 5 6
Measures of Error in Regression

25. Measures of Error in Regression
The linear equation isn’t the whole thing
Also need a measure of error
Three common measures of error
r2 (r-squared)
Sum of the squared residuals = SSresidual= SSerror
Standard error of estimate
Measures of Error in Regression

26. Measures of Error in Regression
R-squared (r2) represents the percent variance in Y accounted for by X
r = 0.8
r2 = 0.64
r = 0.5
r2 = 0.25 Y X 1 2 3 4 5 6
64% of the variance in Y
is explained by X Y X 1 2 3 4 5 6
25% of the variance in Y
is explained by X
Measures of Error in Regression

27. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror Y X 1 2 3 4 5 6
Compute the difference between the predicted values and the observed values (“residuals”)
Square the differences
Add up the squared differences
Measures of Error in Regression

28. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror
X Y
mean
3.6
Predicted values of Y (points on the line)
4.0
Measures of Error in Regression

29. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror
X Y
6.2
= (0.92)(6)+0.688
mean
3.6
Predicted values of Y (points on the line)
4.0
Measures of Error in Regression

30. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror
X Y
6.2
= (0.92)(6)+0.688
1.6
= (0.92)(1)+0.688
5.3
= (0.92)(5)+0.688
3.45
= (0.92)(3)+0.688
3.45
= (0.92)(3)+0.688
mean
3.6
4.0
Measures of Error in Regression

31. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror
X Y Y X 1 2 3 4 5 6
6.2
6.2
1.6
5.3
3.45
1.6
5.3
3.45
3.45
Measures of Error in Regression

32. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror
residuals
X Y
6.2 =
-0.20
1.6 =
0.40
5.3 =
0.70
3.45 =
0.55
3.45
-1.45 =
mean
3.6
4.0
Quick check
0.00
Measures of Error in Regression

33. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror
X Y
6.2
-0.20
0.04
1.6
0.40
0.16
5.3
0.70
0.49
3.45
0.55
0.30
3.45
-1.45
2.10
mean
3.6
4.0
0.00
3.09
SSERROR
Measures of Error in Regression

34. Measures of Error in Regression
Sum of the squared residuals = SSresidual = SSerror
4.0
0.0
16.0
SSY
X Y
6.2
-0.20
0.04
1.6
0.40
0.16
5.3
0.70
0.49
3.45
0.55
0.30
3.45
-1.45
2.10
mean
3.6
4.0
0.00
3.09
SSERROR
Measures of Error in Regression

35. Measures of Error in Regression
Standard error of the estimate represents the average deviation from the line Y X 1 2 3 4 5 6
df = n - 2
Measures of Error in Regression

36. Measures of Error in Regression
SPSS Regression output gives you a lot of stuff r2
percent variance in Y accounted for by X
Standard error of the estimate
the average deviation from the line
SSresiduals or SSerror
Measures of Error in Regression

37. You’ll practice computing the regression equation and error for the “best fitting line” (by hand and using SPSS)
In lab